![]() ![]() So the particle has a zero probability of being found at points other than those where x=na. ![]() If the wavefunction is ψ(x) then the probability density function for the particle position is |ψ(x)|². This is a wavefunction that is concentrated at multiples of some quantity a, ie. Now consider a particle whose wavefunction takes the form of the Dirac comb: In practice there are shortcuts to achieving much the same effect. But if you compute the Fourier transform of a polygonal image, remove suitable high frequency components, and then take the inverse Fourier transform before sampling you'll produce an image that's much more pleasing to the eye. It seems like jaggies have nothing to do with the world of Fourier transforms. The "jaggies" you get from rendering polygons are an example of this phenomenon. Ray tracers will send out many rays for each pixel, in effect forming a much higher resolution image than the resolution of the final result, and that high resolution image is filtered before being sampled down to the final resulting image. In the 3D rendering world you need to do something similar. This is frequently carried out with an analogue filter. In the audio world you need to filter your sound to remove the high frequencies before you sample. Here's what a fraction of a second of music might look like when the pressure of the sound wave is plotted against time: Maybe surprisingly, the worlds of audio and graphics can help us answer this question. Is there a wavefunction that allows us to know the digits to the right of the decimal point as far as we want for both position and momentum measurements? It trivially satisfies Heisenberg's inequality because the variance of the position and the momentum aren't even finite quantities.īut being compatible with Heisenberg uncertainty isn't enough for something to be realisable as a physical state. In other words, it's compatible with the uncertainty principle that we could know the digits beyond the decimal point to as much accuracy as we like as long as we don't know the digits before the point. For example, the following state of affairs is also compatible with the uncertainty principle, in suitably chosen units: ![]()
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